$L_1 ,L_2$ are FDA prove $L_1 \bigotimes L_2$ is FDA.

Question

Let $L_1$ and $L_2$ be FDA (i.e it's possible to build FDA for them). Let $L_1 \bigotimes L_2$ be the language of all that have prefix in $L_1$ and also have prefix in $L_2$, prove that $L_1 \bigotimes L_2$ is FDA.

My Attempt:

I tried somehow to build an automata like product automata:

$A=(Q,\Sigma,q_0,\delta,F)$

$Q=Q_1 \times Q_2$

$q_0=(q_{01},q_{02})$

$\forall q_1\in Q_1, q_2\in Q_2,\delta((q_1,q_2),\sigma)=(\delta_1(q_1,\sigma),\delta_2(q_2,\sigma))$

$F=F_1\times F_2$

Is it possible to modify a bit the paoduct automata for proving it? how? any other suggesions?

Answer 1

First of all, it is not a good idea to introduce your own terminology. It actually may even prevent you to find the solution of your exercise. Your languages $L_1$ and $L_2$ are regular.

Hint. The language of all words having a prefix in $L$ is $L\Sigma^*$. Now use the closure properties of regular languages to solve your exercise.